Bounding ramification by covers and curves

Abstract

We prove that Q ¯ ℓ \bar {\mathbb {Q}}_\ell -local systems of bounded rank and ramification on a smooth variety X X defined over an algebraically closed field k k of characteristic p ≠ ℓ p\neq \ell are tamified outside of codimension 2 2 by a finite separable cover of bounded degree. In rank one, there is a curve which preserves their monodromy. There is a curve defined over the algebraic closure of a purely transcendental extension of k k of finite degree which fulfills the Lefschetz theorem.